Switched differential algebraic equations: Jumps and impulses Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Seminar at Systems, Control and Applied Analysis group University of Groningen, The Netherlands, 2012/02/20
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Contents Introduction 1 Distributions as solutions 2 Review: classical distribution theory Restriction of distributions Piecewise smooth distributions Regularity of matrix pairs and solution formulas 3 Regularity and the quasi-Weierstrass form Solution properties: the homogenous case Solution properties: the inhomogenous case Conclusions 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Standard modeling of circuits i L i L + u C u L C L − d t i L = − 1 d L u C d t i L = 1 d L u d t u C = 1 d C i L General form: x = Ax + Bu ˙ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Switched ODE? i L d t i L = 1 d Mode 1: L u u L L + d t i L = − 1 d Mode 2: L u C u C u C − d t u C = 1 d C i L i C No switched ODE Not possible to write as x ( t ) = A σ ( t ) x + B σ ( t ) u . ˙ Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Include algebraic equations in description i L With x := ( i L , u L , i C , u C ) write each mode as: u L L E p ˙ x = A p x + B p u + u u C C − i C ⇒ Algebraic equations E p singular L d d t i L = u L , C d Mode 1: d t u C = i C , 0 = u L − u , 0 = i C L 0 0 0 0 1 0 0 0 0 0 0 C 0 0 1 0 0 ˙ x = x + u 0 0 0 0 0 1 0 0 − 1 0 0 0 0 0 0 1 0 0 L d d t i L = u L , C d Mode 2: d t u C = i C , 0 = i L − i C , 0 = u L + u C Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Switched DAEs DAE = Differential algebraic equation Switched DAE E σ ( t ) ˙ x ( t ) = A σ ( t ) x ( t ) + B σ ( t ) u ( t ) ( swDAE ) or short E σ ˙ x = A σ x + B σ u with switching signal σ : R → { 1 , 2 , . . . , p } piecewise constant locally finitely many jumps modes ( E 1 , A 1 , B 1 ) , . . . , ( E p , A p , B p ) E p , A p ∈ R n × n , p = 1 , . . . , p B p : R n × m , p = 1 , . . . , p input u : R → R m Question Existence and nature of solutions? Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Impulse example i L i L + + u L u L u u L L − − L d inductivity law: d t i L = u L switch dependent: 0 = u L − u 0 = i Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Impulse example i L i L + + u L u L u u L L − − x = [ i L , u L ] ⊤ x = [ i L , u L ] ⊤ � 0 � L � � 0 � � � L � � 0 � � 0 � 0 1 0 1 x = ˙ x + u x = ˙ x + u 0 0 0 1 − 1 0 0 1 0 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Solution of example L d d t i L = u L , 0 = u L − u or 0 = i L Assume: u constant, i L (0) = 0 � 1 , t < t s switch at t s > 0: σ ( t ) = 2 , t ≥ t s u L ( t ) i L ( t ) u t t t s t s δ t s Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Contents Introduction 1 Distributions as solutions 2 Review: classical distribution theory Restriction of distributions Piecewise smooth distributions Regularity of matrix pairs and solution formulas 3 Regularity and the quasi-Weierstrass form Solution properties: the homogenous case Solution properties: the inhomogenous case Conclusions 4 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Distribution theorie - basic ideas Distributions - overview Generalized functions Arbitrarily often differentiable Dirac-Impulse δ 0 is “derivative” of Heaviside step function ✶ [0 , ∞ ) Two different formal approaches Functional analytical: Dual space of the space of test functions 1 (L. Schwartz 1950) Axiomatic: Space of all “derivatives” of continuous functions 2 (J. Sebasti˜ ao e Silva 1954) Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Distributions - formal Definition (Test functions) C ∞ := { ϕ : R → R | ϕ is smooth with compact support } 0 Definition (Distributions) D := { D : C ∞ → R | D is linear and continuous } 0 Definition (Regular distributions) f D : C ∞ � f ∈ L 1 , loc ( R → R ): → R , ϕ �→ R f ( t ) ϕ ( t )d t ∈ D 0 Dirac Impulse at t 0 ∈ R Definition (Derivative) δ t 0 : C ∞ D ′ ( ϕ ) := − D ( ϕ ′ ) → R , ϕ �→ ϕ ( t 0 ) 0 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Multiplication with functions Definition (Multiplication with smooth functions) α ∈ C ∞ : ( α D )( ϕ ) := D ( αϕ ) (swDAE) E σ ˙ x = A σ x + B σ u Coefficients not smooth ∈ C ∞ Problem: E σ , A σ , B σ / Observation: E σ ˙ x = A σ x + B σ u ⇔ ∀ i ∈ Z : ( E p i ˙ x ) [ t i , t i +1 ) = ( A p i x + B p i u ) [ t i , t i +1 ) i ∈ Z : σ [ t i , t i +1 ) ≡ p i New question: Restriction of distributions Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Desired properties of distributional restriction Distributional restriction: { M ⊆ R | M interval } × D → D , ( M , D ) �→ D M and for each interval M ⊆ R D �→ D M is a projection (linear and idempotent) 1 ∀ f ∈ L 1 , loc : ( f D ) M = ( f M ) D 2 � � supp ϕ ⊆ M ⇒ D M ( ϕ ) = D ( ϕ ) ∀ ϕ ∈ C ∞ : 3 0 supp ϕ ∩ M = ∅ ⇒ D M ( ϕ ) = 0 ( M i ) i ∈ N pairwise disjoint, M = � i ∈ N M i : 4 � D M = D M i , D M 1 ˙ ∪ M 2 = D M 1 + D M 2 , ( D M 1 ) M 2 = 0 i ∈ N Theorem Such a distributional restriction does not exist. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Proof of non-existence of restriction Consider the following distribution(!): d i := ( − 1) i � D := d i δ d i , i + 1 i ∈ N - 1 - 1 2 4 0 1 1 3 Restriction should give � D (0 , ∞ ) = d 2 k δ d 2 k k ∈ N Choose ϕ ∈ C ∞ such that ϕ [0 , 1] ≡ 1: 0 1 � � D (0 , ∞ ) ( ϕ ) = d 2 k = 2 k + 1 = ∞ k ∈ N k ∈ N Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Dilemma Distributions Switched DAEs Distributional restriction not Examples: distributional possible solutions Multiplication with non-smooth Multiplication with non-smooth coefficients not possible coefficients Initial value problems cannot be Or: Restriction on intervals formulated Underlying problem Space of distributions too big. Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
Introduction Distributions as solutions Regularity & Solution formulas Conclusions Piecewise smooth distributions Define a suitable smaller space: Definition (Piecewise smooth distributions D pw C ∞ ) f ∈ C ∞ � pw , � � D pw C ∞ := � T ⊆ R locally finite , f D + D t � i δ ( i ) � ∀ t ∈ T : D t = � n t i =0 a t t ∈ T � t f D D t i − 1 D t i +1 D t i t i − 1 t i t i +1 Stephan Trenn Technomathematics group, University of Kaiserslautern, Germany Switched differential algebraic equations:Jumps and impulses
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